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All Textbook Solutions for Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
In Problems 1-8, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and the initial simplex tableau. (C) Find the optimal solution of the modified problem by applying the simplex method to the initial simplex tableau. (D) Find the optimal solution of the original problem, if it exists. MaximizeP=5x1+2x2subjecttox1+2x212x1+x24x1,x20In Problems 1-8, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and the initial simplex tableau. (C) Find the optimal solution of the modified problem by applying the simplex method to the initial simplex tableau. (D) Find the optimal solution of the original problem, if it exists. MaximizeP=3x1+7x2subjectto2x1+x216x1+x26x1,x20In Problems 1-8, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and the initial simplex tableau. (C) Find the optimal solution of the modified problem by applying the simplex method to the initial simplex tableau. (D) Find the optimal solution of the original problem, if it exists. MaximizeP=3x1+5x2subjectto2x1+x28x1+x2=6x1,x20In Problems 1-8, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and the initial simplex tableau. (C) Find the optimal solution of the modified problem by applying the simplex method to the initial simplex tableau. (D) Find the optimal solution of the original problem, if it exists. MaximizeP=4x1+3x2subjecttox1+3x224x1+x2=12x1,x20In Problems 1-8, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and the initial simplex tableau. (C) Find the optimal solution of the modified problem by applying the simplex method to the initial simplex tableau. (D) Find the optimal solution of the original problem, if it exists. MaximizeP=4x1+3x2subjecttox1+2x22x1+x24x1,x20In Problems 1-8, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and the initial simplex tableau. (C) Find the optimal solution of the modified problem by applying the simplex method to the initial simplex tableau. (D) Find the optimal solution of the original problem, if it exists. MaximizeP=3x1+4x2subjecttox12x22x1+x25x1,x20In Problems 1-8, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and the initial simplex tableau. (C) Find the optimal solution of the modified problem by applying the simplex method to the initial simplex tableau. (D) Find the optimal solution of the original problem, if it exists. MaximizeP=5x1+10x2subjecttox1+x232x1+3x212x1,x20In Problems 1-8, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and the initial simplex tableau. (C) Find the optimal solution of the modified problem by applying the simplex method to the initial simplex tableau. (D) Find the optimal solution of the original problem, if it exists. MaximizeP=4x1+6x2subjecttox1+x223x1+5x215x1,x20Use the big M method to solve Problems 9-22. MinimizeandmaximizeP=2x1x2subjecttox1+x285x1+3x230x1,x20Use the big M method to solve Problems 9-22. MinimizeandmaximizeP=4x1+16x2subjectto3x1+x228x1+2x216x1,x20Use the big M method to solve Problems 9-22. MinimizeP=2x1+5x2subjecttox1+2x2182x1+x221x1+x210x1,x20Use the big M method to solve Problems 9-22. MaximizeP=6x1+2x2subjecttox1+2x2202x1+x216x1+x29x1,x20Use the big M method to solve Problems 9-22. MaximizeP=10x1+12x2+20x3subjectto3x1+x2+2x312x1x2+2x3=6x1,x2,x30Use the big M method to solve Problems 9-22. MaximizeP=5x1+7x2+9x3subjecttox1x2+x3202x1+x2+5x3=35x1,x2,x30Use the big M method to solve Problems 9-22. MinimizeC=5x112x2+16x3subjecttox1+2x2+x3102x1+3x2+x362x1+x2x3=1x1,x2,x30Use the big M method to solve Problems 9-22. MinimizeC=3x1+15x24x3subjectto2x1+x2+3x324x1+2x2+x36x13x2+x3=2x1,x2,x30Use the big M method to solve Problems 9-22. MazimizeP=3x1+5x2+6x3subjectto2x1+x2+2x382x1+x22x3=0x1,x2,x30Use the big M method to solve Problems 9-22. MazimizeP=3x1+6x2+2x3subjectto2x1+2x2+3x3122x12x2+x3=0x1,x2,x30Use the big M method to solve Problems 9-22. MaximizeP=2x1+3x2+4x3subjecttox1+2x2+x3252x1+x2+2x360x1+2x2x310x1,x2,x30Use the big M method to solve Problems 9-22. MaximizeP=5x1+2x2+9x3subjectto2x1+4x2+x31503x1+3x2+x390x1+5x2+x3120x1,x2,x30Use the big M method to solve Problems 9-22. MaximizeP=x1+2x2+5x3subjecttox1+3x2+2x3602x1+5x2+2x350x12x2+x340x1,x2,x30Use the big M method to solve Problems 9-22. MaximizeP=2x1+4x2+x3subjectto2x1+3x2+5x32802x1+2x2+x31402x1+x2+150x1,x2,x30Solve Problems 5 and 7 by graphing (the geometric method).Solve Problems 6 and 8 by graphing (the geometric method).Problems 25-32 are mixed. Some can be solved by the methods presented in Sections 6.2 and 6.3 while others must be solved by the big M method. MinimizeC=10x140x25x3subjecttox1+3x264x2+x33x1,x2,x30Problems 25-32 are mixed. Some can be solved by the methods presented in Sections 6.2 and 6.3 while others must be solved by the big M method. MaximizeP=7x15x2+2x3subjecttox12x2+x38x1x2+x310x1,x2,x30Problems 25-32 are mixed. Some can be solved by the methods presented in Sections 6.2 and 6.3 while others must be solved by the big M method. MaximizeP=5x1+10x2+15x3subjectto2x1+3x2+x324x12x22x31x1,x2,x30Problems 25-32 are mixed. Some can be solved by the methods presented in Sections 6.2 and 6.3 while others must be solved by the big M method. MinimizeC=5x1+10x2+15x3subjectto2x1+3x2+x324x12x22x31x1,x2,x30Problems 25-32 are mixed. Some can be solved by the methods presented in Sections 6.2 and 6.3 while others must be solved by the big M method. MinimizeC=10x1+40x2+5x3subjecttox1+3x264x2+3x33x1,x2,x30Problems 25-32 are mixed. Some can be solved by the methods presented in Sections 6.2 and 6.3 while others must be solved by the big M method. MaximizeP=8x1+2x210x3subjecttox1+x23x364x1x2+2x37x1,x2,x30Problems 25-32 are mixed. Some can be solved by the methods presented in Sections 6.2 and 6.3 while others must be solved by the big M method. MaximizeP=12x1+9x2+5x3subjecttox1+3x2+x3402x1+x2+3x360x1,x2,x30Problems 25-32 are mixed. Some can be solved by the methods presented in Sections 6.2 and 6.3 while others must be solved by the big M method. MinimizeC=10x1+12x2+28x3subjectto4x1+2x2+3x3202x1x24x310x1,x2,x30In Problems 33-38, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model). Then solve the problem using the big M method. Advertising. An advertising company wants to attract new customers by placing a total of at most 10 ads in 3 newspapers. Each ad in the Sentinel costs $200 and will be read by 2,000 people. Each ad in the Journal costs $200 and will be read by 500 people. Each ad in the Tribune costs $100 and will be read by 1,500 people. The company wants at least 16,000 people to read its ads. How many ads should it place in each paper in order to minimize the advertising costs? What is the minimum cost?In Problems 33-38, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model). Then solve the problem using the big M method. Advertising. Discuss the effect on the solution to Problem 33 if the Tribune will not accept more than 4 ads from the company.In Problems 33-38, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model). Then solve the problem using the big M method. Human nutrition. A person on a high-protein, low-carbohydrate diet requires at least 100 units of protein and at most 24 units of carbohydrates daily. The diet will consist entirely of three special liquid diet foods: A,B, and C. The contents and costs of the diet foods are given in the table. How many bottles of each brand of diet food should be consumed daily in order to meet the protein and carbohydrate requirements at minimal cost? What is the minimum cost?In Problems 33-38, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model). Then solve the problem using the big M method. Human nutrition. Discuss the effect on the solution to Problem 35 if the cost of brand C liquid diet food increases to $1.50 per bottle.In Problems 33-38, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model). Then solve the problem using the big M method. Plant food. A farmer can use three types of plant food: mix A, mix B, and mix C. The amounts (in pounds) of nitrogen, phosphoric acid, and potash in cubic yard of each mix are given in the table. Tests performed on the soil indicate that the field needs at least 800 pounds of potash. The tests also indicate that no more than 700 pounds of phosphoric acid should be added to the field. The farmer plans to plant a crop that requires a great deal of nitrogen. How many cubic yards of each mix should be added to the field in order to satisfy the potash and phosphoric acid requirements and maximize the amount of nitrogen? What is the maximum amount of nitrogen?In Problems 33-38, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model). Then solve the problem using the big M method. Plant food. Discuss the effect on the solution to Problem 37 if the limit on phosphoric acid is increased to 1,000 pounds.In Problems 39-47, construct a mathematical model in the form of a linear programming problem. Do not solve. Manufacturing. A company manufactures car and truck frames at plants in Milwaukee and Racine. The Milwaukee plant has a daily operating budget of $50,000 and can produce at most 300 frames daily in any combination. It costs $150 to manufacture a car frame and $200 to manufacture a truck frame at the Milwaukee plant. The Racine plant has a daily operating budget of $35,000, and can produce a maximum combined total of 200 frames daily. It costs $135 to manufacture a car frame and $180 to manufacture a truck frame at the Racine plant. Based on past demand, the company wants to limit production to a maximum of 250 car frames and 350 truck frames per day. If the company realizes a profit of $50 on each car frame and $70 on each truck frame, how many frames of each type should be produced at each plant to maximize the daily profit?In Problems 39-47, construct a mathematical model in the form of a linear programming problem. Do not solve. Loan distributions. A savings and loan company has $3 million to lend. The types of loans and annual returns offered are given in the table. State laws require that at least 50% of the money loaned for mortgages must be for first mortgages and that at least 30% of the total amount loaned must be for either first or second mortgages. Company policy requires that the amount of signature and automobile loans cannot exceed 25% of the total amount loaned and that signature loans cannot exceed 15% of the total amount loaned. How much money should be allocated to each type of loan in order to maximize the company’s return?In Problems 39-47, construct a mathematical model in the form of a linear programming problem. Do not solve. Oil refining. A refinery produces two grades of gasoline, regular and premium, by blending together three components: A,B, and C. Component A has an octane rating of 90 and costs $28 a barrel, component B has an octane rating of 100 and costs $30 a barrel, and component C has an octane rating of 110 and costs $34 a barrel. The octane rating for regular must be at least 95 and the octane rating for premium must be at least 105. Regular gasoline sells for $38 a barrel and premium sells for $46 a barrel. The company has 40,000 barrels of component A, 25000 barrels of component B, and 15,000 barrels of component C. It must produce at least 30,000 barrels of regular and 25,000 barrels of premium. How should the components be blended in order to maximize profit?In Problems 39-47, construct a mathematical model in the form of a linear programming problem. Do not solve. Trail mix. A company makes two brands of trail mix, regular and deluxe, by mixing dried fruits, nuts, and cereal. The recipes for the mixes are given in the table. The company has 1,200 pounds of dried fruits, 750 pounds of nuts, and 1,500 pounds of cereal for the mixes. The company makes a profit of $0.40 on each pound of regular mix and $0.70 on each pound of deluxe mix. How many pounds of each ingredient should be used in each mix in order to maximize the company’s profit?In Problems 39-47, construct a mathematical model in the form of a linear programming problem. Do not solve. Investment strategy. An investor is planning to divide her investments among high-tech mutual funds, global mutual funds, corporate bonds, municipal bonds, and CDs. Each of these investments has an estimated annual return and a risk factor (see the table). The risk level for each choice is the product of its risk factor and the percentage of the total funds invested in that choice. The total risk level is the sum of the risk levels for all the investments. The investor wants at least 20% of her investments to be in CDs and does not want the risk level to exceed 1.8. What percentage of her total investments should be invested in each choice to maximize the return?In Problems 39-47, construct a mathematical model in the form of a linear programming problem. Do not solve. Investment strategy. Refer to Problem 43. Suppose the investor decides that she would like to minimize the total risk factor, as long as her return does not fall below 9%. What percentage of her total investments should be invested in each choice to minimize the total risk level?In Problems 39-47, construct a mathematical model in the form of a linear programming problem. Do not solve. Human nutrition. A dietician arranges a special diet using foods L,M, and N. The table gives the nutritional contents and cost of 1 ounce of each food. The diet’s daily requirements are at least 400 units of calcium, at least 200 units of iron, at least 300 units of vitamin A, at most 150 units of cholesterol, and at most 900 calories. How many ounces of each food should be used in order to meet the diet’s requirements at a minimal cost?In Problems 39-47, construct a mathematical model in the form of a linear programming problem. Do not solve. Mixing feed. A farmer grows three crops: corn, oats, and soybeans. He mixes them to feed his cows and pigs. At least 40% of the feed mix for the cows must be corn. The feed mix for the pigs must contain at least twice as much soybeans as corn. He has harvested 1,000 bushels of corn. 500 bushels of oats, and 1,000 bushels of soybeans. He needs 1,000 bushels of each feed mix for his livestock. The unused corn, oats, and soybeans can be sold for $4,$3.50, and $3.25 a bushel, respectively (thus, these amounts also represent the cost of the crops used to feed the livestock). How many bushels of each crop should be used in each feed mix in order to produce sufficient food for the livestock at a minimal cost?In Problems 39-47, construct a mathematical model in the form of a linear programming problem. Do not solve. Transportation. Three towns are forming a consolidated school district with two high schools. Each high school has a maximum capacity of 2,000 students. Town A has 500 high school students, town B has 1,200, and town C has 1,800. The weekly costs of transporting a student from each town to each school are given in the table. In order to balance the enrollment, the school board decided that each high school must enroll at least 40% of the total student population. Furthermore, no more than 60% of the students in any town should be sent to the same high school. How many students from each town should be enrolled in each school in order to meet these requirements and minimize the cost of transporting the students?Problems 1-7 refer to the partially completed table of the six basic solutions of the e -system 2x1+5x2+s1=32x1+2x2+s2=14x1x2s1s2A003214B06.401.2C0730D16002E00F00 In basic solution B, which variables are basic?Problems 1-7 refer to the partially completed table of the six basic solutions of the e -system 2x1+5x2+s1=32x1+2x2+s2=14x1x2s1s2A003214B06.401.2C0730D16002E00F00 In basic solution D, which variables are nonbasic?Problems 1-7 refer to the partially completed table of the six basic solutions of the e -system 2x1+5x2+s1=32x1+2x2+s2=14x1x2s1s2A003214B06.401.2C0730D16002E00F00 Find basic solution E.Problems 1-7 refer to the partially completed table of the six basic solutions of the e -system 2x1+5x2+s1=32x1+2x2+s2=14x1x2s1s2A003214B06.401.2C0730D16002E00F00 Find basic solution F.Problems 1-7 refer to the partially completed table of the six basic solutions of the e -system 2x1+5x2+s1=32x1+2x2+s2=14x1x2s1s2A003214B06.401.2C0730D16002E00F00 Which of the six basic solutions are feasible?Problems 1-7 refer to the partially completed table of the six basic solutions of the e -system 2x1+5x2+s1=32x1+2x2+s2=14x1x2s1s2A003214B06.401.2C0730D16002E00F00 Describe geometrically the set of points in the plane such that s10.Problems 1-7 refer to the partially completed table of the six basic solutions of the e -system 2x1+5x2+s1=32x1+2x2+s2=14x1x2s1s2A003214B06.401.2C0730D16002E00F00 Use the basic feasible solutions to maximize P=50x1+60x2.A linear programming problem has 6 decision variables and 3 problem constraints. How many rows are there in the table of basic solutions of the corresponding e -system?Given the linear programming problem MaximizeP=6x1+2x2subjectto2x1+x28x1+2x210x1,x20 convert the problem constraints into a system of equations using the slack variables.How many basic variables and how many nonbasic variables are associated with the system in Problem 9?Find all basic solutions for the system in Problem 9, and determine which basic solutions are feasible.Write the simplex tableau for Problem 9, and circle the pivot element. Indicate the entering and exiting variables.Solve Problem 9 using the simplex method.For the simplex tableau below, identify the basic and non-basic variables. Find the pivot element, the entering and exiting variables, and perform one pivot operation. x1x2x3s1s2s3P213100020304110030205201010805300150Find the basic solution for each tableau. Determine whether the optimal solution has been reached, additional pivoting is required, or the problem has no optimal solution. Ax1x2s1s2P4100022011052030112Bx1x2s1s2P1301070210002100122Cx1x2s1s2P12040602160150302110Form the dual problem of MinimizeC=5x1+2x2subjecttox1+3x2152x1+x220x1,x20Write the initial system for the dual problem in Problem 16.Write the first simplex tableau for the dual problem in Problem 16 and label the columns.Use the simplex method to find the optimal solution of the dual problem in Problem 16.Use the final simplex tableau from Problem 19 to find the optimal solution of the linear programming problem in Problem 16.Solve the linear programming problem using the simplex method. MaximizeP=3x1+4x2subjectto2x1+4x2243x1+3x2214x1+2x220x1,x20Form the dual problem of the linear programming problem MinimizeC=3x1+8x2subjecttox1+x210x1+2x215x23x1,x20Solve Problem 22 by applying the simplex method to the dual problem.Solve the linear programming Problems 24 and 25. MaximizeP=5x1+3x23x3subjecttox1x22x332x1+2x25x310x1,x2,x30Solve the linear programming Problems 24 and 25. MaximizeP=5x1+3x23x3subjecttox1x22x33x1+x25x1,x2,x30Solve the linear programming problem using the table method: MaximizeP=10x1+7x2+8x3subjectto2x1+x2+3x312x1,x20Refer to Problem 26. How many pivot columns are required to solve the linear programming problem using the simplex method?In problems 28 and 29, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and find the initial simplex tableau. (C) Find the optimal solution of the modified problem by applying the simplex method to the initial simplex tableau. (D) Find the optimal solution of the original problem, if it exists. MaximizeP=x1+3x2subjecttox1+x26x1+2x28x1,x20In problems 28 and 29, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and find the initial simplex tableau. (C) Find the optimal solution of the modified problem by applying the simplex method to the initial simplex tableau. (D) Find the optimal solution of the original problem, if it exists. MaximizeP=x1+x2subjecttox1+x25x1+2x24x1,x20Find the modified problem for the following linear programming problem. (Do not solve.) MaximizeP=2x1+3x2+x3subjecttox13x2+x37x1x2+2x323x1+2x2x3=4x1,x2,x30Write a brief verbal description of the type of linear programming problem that can be solved by the method indicated in Problems 31-33. Include the type of optimization, the number of variables, the type of constraints, and any restrictions on the coefficients and constants. Basic simplex method with slack variablesWrite a brief verbal description of the type of linear programming problem that can be solved by the method indicated in Problems 31-33. Include the type of optimization, the number of variables, the type of constraints, and any restrictions on the coefficients and constants. Dual problem methodWrite a brief verbal description of the type of linear programming problem that can be solved by the method indicated in Problems 31-33. Include the type of optimization, the number of variables, the type of constraints, and any restrictions on the coefficients and constants. Big M methodSolve the following linear programming problem by the simplex method, keeping track of the obvious basic solution at each step. Then graph the feasible region and illustrate the path to the optimal solution determined by the simplex method. MaximizeP=2x1+3x2subjecttox1+2x2223x1+x226x18x210x1,x20Solve by the dual problem method: MinimizeC=3x1+2x2subjectto2x1+x2202x1+x29x1+x26x1,x20Solve Problem 35 by the big M method.Solve by the dual problem method: MinimizeC=15x1+12x2+15x3+18x4subjecttox1+x2240x3+x4500x1+x3400x2+x4300x1,x2,x3,x40In problems 38-41, construct a mathematical model in the form of a linear programming problem (The answer in the back of the book for these application problems include the model.) Then solve the problem by the simplex, dual problem, or big M methods. Investment. An investor has $150,000 to invest in oil stock, steel stock, and government bonds. The bonds are guaranteed to yield 5%, but the yield for each stock can vary. To protect against major losses, the investor decides that the amount invested in oil stock should not exceed $50,000 and that the total amount invested in stock cannot exceed the amount invested in bonds by more than $25,000. (A) If the oil stock yields 12% and the steel stock yields 9%, how much money should be invested in each alternative in order to maximize the return? What is the maximum return? (B) Repeat part (A) if the oil stock yields 9% and the steel stock yields 12%.In problems 38-41, construct a mathematical model in the form of a linear programming problem (The answer in the back of the book for these application problems include the model.) Then solve the problem by the simplex, dual problem, or big M methods. Manufacturing. A company manufactures outdoor furniture consisting of regular chairs, rocking chairs, and chaise lounges. Each piece of furniture passes through three different production departments: fabrication, assembly, and finishing. Each regular chair takes 1 hour to fabricate, 2 hours to assemble, and 3 hours to finish. Each rocking chair takes 2 hours to fabricate, 2 hours to assemble, and 3 to finish. Each chaise lounge takes 3 hours to fabricate, 4 hours to assemble, and 2 hours to finish. There are 2,500 labor-hours available in the fabrication department, 3,000 in the assembly department, and 3,500 in the finishing department. The company makes a profit of $17 on each regular chair, $24 on each rocking chair, and $31 on each chaise lounge. (A) How many chairs of each type should the company produce in order to maximize profit? What is the maximum profit? (B) Discuss the effect on the optimal solution in part (A) if the profit on a regular chair is increased to $25 and all other data remain the same. (C) Discuss the effect on the optimal solution in part (A) if the available hours on the finishing department are reduced to 3,000 and all other data remain the same.In problems 38-41, construct a mathematical model in the form of a linear programming problem (The answer in the back of the book for these application problems include the model.) Then solve the problem by the simplex, dual problem, or big M methods. Shipping schedules. A company produces motors for washing machines at factory A and factory B. The motors are then shipped to either plant X or plant Y, where the washing machines are assembled. The maximum number of motors that can be produced at each factory monthly, the minimum number required for each plant to meet anticipated demand, and the shipping charges for one motor are given in the table. Determine a shipping schedule that will minimize the cost of transporting the motors from the factories to the assembly plants. PlantXPlantYMaximumProductionFactoryA$5$81,500FactoryB$9$71,000MinimumRequirement9001,200In problems 38-41, construct a mathematical model in the form of a linear programming problem (The answer in the back of the book for these application problems include the model.) Then solve the problem by the simplex, dual problem, or big M methods. Blending-food processing. A company blends long-grain rice and wild rice to produce two brands of rice mixes: brand A, which is marketed under the company’s name, and brand B, which is marketed as a generic brand. Brand A must contain at least 10% wild rice, and brand B must contain at least 5% wild rice. Long-grain rice costs $0.70 per pound, and wild rice costs $3.40 per pound. The company sells brand A for $1.50 a pound and brand B for $1.20 a pound. The company has 8,000 pounds of long-grain rice and 500 pounds of wild rice on hand. How should the company use the available rice to maximize its profit? What is the maximum profit?If a compound proposition contains three variables p,q,andr, then its truth table will have eight rows, one for each of the eight ways of assigning truth values to p,q,andrTTT,TTF,TFT,TFF,FTT,FTF,FFT,FFF. Construct truth tables to verify the following logical implications and equivalences: ApqqrprBpqrpqprCpqrpqprConsider the propositions p and q : p:142200q:232500 Express each of the following propositions as an English sentence and determine whether it is true or false. ApBqCpqDpqEpqConsider the propositions pandq : p:52+122=132q:72+242=252 Express each of the following propositions as an English sentence and determine whether it is true or false. ApqBTheconverseofpqAThecontrapositiveofpqConstruct truth table for pq.Construct the truth table for pqqp.Construct the truth table for pqpq.Show that pqpqp.Show that pqpq.Use equivalences from Table 2 to show that pqqp.In Problems 1-6, refer to the footnote for the definitions of divisors, multiple, prime, even, and odd.* List the positive integers that are divisors of 20. *An integer d is a divisor of an integer n (and n is a multiple of d ) if n=kd for some integer for some integer k. An integer n is even if 2 is a divisor of n ; otherwise, n is odd. An integer p1 is prime if its only positive divisors are 1 and p.In Problems 1-6, refer to the footnote for the definitions of divisors, multiple, prime, even, and odd.* List the positive integers that are divisors of 24. *An integer d is a divisor of an integer n (and n is a multiple of d ) if n=kd for some integer for some integer k. An integer n is even if 2 is a divisor of n ; otherwise, n is odd. An integer p1 is prime if its only positive divisors are 1 and p.In Problems 1-6, refer to the footnote for the definitions of divisors, multiple, prime, even, and odd.* List the positive multiples of 11 that are less than 60. *An integer d is a divisor of an integer n (and n is a multiple of d ) if n=kd for some integer for some integer k. An integer n is even if 2 is a divisor of n ; otherwise, n is odd. An integer p1 is prime if its only positive divisors are 1 and p.In Problems 1-6, refer to the footnote for the definitions of divisors, multiple, prime, even, and odd.* List the positive multiples of 9 that are less than 50. *An integer d is a divisor of an integer n (and n is a multiple of d ) if n=kd for some integer for some integer k. An integer n is even if 2 is a divisor of n ; otherwise, n is odd. An integer p1 is prime if its only positive divisors are 1 and p.In Problems 1-6, refer to the footnote for the definitions of divisors, multiple, prime, even, and odd.* List the primes between 20 and 30. *An integer d is a divisor of an integer n (and n is a multiple of d ) if n=kd for some integer for some integer k. An integer n is even if 2 is a divisor of n ; otherwise, n is odd. An integer p1 is prime if its only positive divisors are 1 and p.In Problems 1-6, refer to the footnote for the definitions of divisors, multiple, prime, even, and odd.* List the primes between 10 and 20. *An integer d is a divisor of an integer n (and n is a multiple of d ) if n=kd for some integer for some integer k. An integer n is even if 2 is a divisor of n ; otherwise, n is odd. An integer p1 is prime if its only positive divisors are 1 and p.Explain why the sum of any two odd integers is even.Explain why the product of any two odd integers is odd.In Problem 9-14, express each proposition as an English sentence and determine whether it is true or false, where p and q are the propositions p:91isprime q:91isodd qIn Problem 9-14, express each proposition as an English sentence and determine whether it is true or false, where p and q are the propositions p:91isprime q:91isodd pqIn Problem 9-14, express each proposition as an English sentence and determine whether it is true or false, where p and q are the propositions p:91isprime q:91isodd pqIn Problem 9-14, express each proposition as an English sentence and determine whether it is true or false, where p and q are the propositions p:91isprime q:91isodd pqIn Problem 9-14, express each proposition as an English sentence and determine whether it is true or false, where p and q are the propositions p:91isprime q:91isodd The converse of pqIn Problem 9-14, express each proposition as an English sentence and determine whether it is true or false, where p and q are the propositions p:91isprime q:91isodd The contrapositive of pqIn Problem 15-20, express each proposition as an English sentence and determine whether it is true or false, where it is true of false, where r and s are the propositions r:themoonisacubes:rainiswet rsIn Problem 15-20, express each proposition as an English sentence and determine whether it is true or false, where it is true of false, where r and s are the propositions r:themoonisacubes:rainiswet rsIn Problem 15-20, express each proposition as an English sentence and determine whether it is true or false, where it is true of false, where r and s are the propositions r:themoonisacubes:rainiswet rsIn Problem 15-20, express each proposition as an English sentence and determine whether it is true or false, where it is true of false, where r and s are the propositions r:themoonisacubes:rainiswet rIn Problem 15-20, express each proposition as an English sentence and determine whether it is true or false, where it is true of false, where r and s are the propositions r:themoonisacubes:rainiswet The contrapositive of rsIn Problem 15-20, express each proposition as an English sentence and determine whether it is true or false, where it is true of false, where r and s are the propositions r:themoonisacubes:rainiswet The converse of rsIn Problem 21-28, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true of false, 30or30In Problem 21-28, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true of false, 30and30In Problem 21-28, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true of false, 30,then320In Problem 21-28, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true of false, 3 is not greater than 0In Problem 21-28, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true of false, 11 is not prime.In Problem 21-28, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true of false, 9 is even or 9 is primeIn Problem 21-28, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true of false, 7 is odd and 7 is primeIn Problem 21-28, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true of false, If 4 is even, then 4 is primeIn Problem 29-34, state the converse and the contrapositive of the given proposition. If triangle ABC is equilateral, then triangle ABC is equiangular.In Problem 29-34, state the converse and the contrapositive of the given proposition. If triangle ABC is isosceles, then the base angles of triangle ABC are congruent.In Problem 29-34, state the converse and the contrapositive of the given proposition. If fx is a linear function with positive slope, then fx is an increasing function.In Problem 29-34, state the converse and the contrapositive of the given proposition. If gx is a quadratic function, then gx is a function that is neither increasing nor decreasing.In Problem 29-34, state the converse and the contrapositive of the given proposition. If n is an integer that is a multiple of 8, then n is an integer that is a multiple of 2 and a multiple of 4.In Problem 29-34, state the converse and the contrapositive of the given proposition. If n is an integer that is a multiple of 6, then n is an integer that is a multiple of 2 and a multiple of 3.In Problem 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pqIn Problem 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pqIn Problem 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pqIn Problem 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pqIn Problem 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. qpqIn Problem 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. qpqIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. ppqIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. ppqIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. ppqIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. ppqIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pqpIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pqqIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. ppqIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. ppqIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. qpqIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. qpqIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pqqpIn Problems 35-52, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pqpqIn Problems 53-58, construct a truth table to verify each implication. ppqIn Problems 53-58, construct a truth table to verify each implication. ppqIn Problems 53-58, construct a truth table to verify each implication. pqpqIn Problems 53-58, construct a truth table to verify each implication. pqpqIn Problems 53-58, construct a truth table to verify each implication. pqqpIn Problems 53-58, construct a truth table to verify each implication. ppqIn Problems 59-64, construct a truth table to verify each equivalence. ppqpqIn Problems 59-64, construct a truth table to verify each equivalence. qpqpqIn Problems 59-64, construct a truth table to verify each equivalence. qpqqpqIn Problems 59-64, construct a truth table to verify each equivalence. ppqpqIn Problems 59-64, construct a truth table to verify each equivalence. ppqppqIn Problems 59-64, construct a truth table to verify each equivalence. ppqpqIn Problems 65-68, verify each equivalence using formulas from Table 2. pqpqIn Problems 65-68, verify each equivalence using formulas from Table 2. pqpqIn Problems 65-68, verify each equivalence using formulas from Table 2. pqpqIn Problems 65-68, verify each equivalence using formulas from Table 2. pqqpIf p is the proposition “I want pickles” and q is the proposition “I want tomatoes,” rewrite the sentence “I want tomatoes, but I do not want pickles” using symbols.Let be the proposition “every politician is honest.” Explain why the statement “every politician is dishonest’ is not equivalence to p. Express p as an English sentence without using word not.If the conditional proposition p is a contingency, is p a contingency, a tautology, or a contradiction? Explain.If the conditional proposition p is a contradiction, is p a contingency, a tautology, or a contradiction? Explain.Can a conditional proposition be false if its contrapositive is true? Explain.Can a conditional proposition be false if its converse is true? Explain.Which of the following statements are true? ABC=0D0In Example 6, find the number of voters in the set CMM. Describe this set verbally and with a Venn diagram.Let G be the set of all numbers such that x2=9. A Denote G by the rule method. B Denote G by the listing method. C Indicate whether the following are true or false: 3G,9G.Given A=0,2,4,6,B=0,1,2,3,4,5,6 and C=2,6,0,4, indicate whether the following relationships are true T or false F : AABBACCA=CDCBEBAFB3MP4MP5MP6MPRefer to Example 7. A How many customers purchased homeowner’s insurance? B Shade the region AL in Figure 10. Find nAL. C Shade the region AHL in Figure 10. Find nAHL.1EIn Problems 1-6, answer yes or no. (If necessary, review Section A.1). Is the set of rational numbers a subset of the set of integers?In Problems 1-6, answer yes or no. (If necessary, review Section A.1). Is the set of integers the intersection of the set of even integers and the set of odd integers?In Problems 1-6, answer yes or no. (If necessary, review Section A.1). Is the set of integers the union of the set of even integers and the set of odd integers?In Problems 1-6, answer yes or no. (If necessary, review Section A.1). If the universal set is the set of integers, is the set of positive integers the complement of the set of negative integers?In Problems 1-6, answer yes or no. (If necessary, review Section A.1). If the universal set is the set of integers, is the set of even integers the complement of the set of odd integers?In Problems 7-14, indicate true T or false F. 1,22,1In Problems 7-14, indicate true T or false F. 3,2,11,2,3,4In Problems 7-14, indicate true T or false F. 5,10=10,5In Problems 7-14, indicate true T or false F. 110,11In Problems 7-14, indicate true T or false F. 00,0In Problems 7-14, indicate true T or false F. 0,6=6In Problems 7-14, indicate true T or false F. 81,2,414EIn Problems 15-28, write the resulting set using the listing method. 1,2,32,3,4In Problems 15-28, write the resulting set using the listing method. 1,2,44,8,16In Problems 15-28, write the resulting set using the listing method. 1,2,32,3,4In Problems 15-28, write the resulting set using the listing method. 1,2,44,8,16In Problems 15-28, write the resulting set using the listing method. 1,4,710,13In Problems 15-28, write the resulting set using the listing method. 3,11,3In Problems 15-28, write the resulting set using the listing method. 1,4,710,1322EIn Problems 15-28, write the resulting set using the listing method. xx2=25In Problems 15-28, write the resulting set using the listing method. xx2=36In Problems 15-28, write the resulting set using the listing method. xx3=27In Problems 15-28, write the resulting set using the listing method. xx4=16In Problems 15-28, write the resulting set using the listing method. xxisanoddnumberbetween1and9,inclusiveIn Problems 15-28, write the resulting set using the listing method. xxisamonthstartingwithMFor U=1,2,3,4,5 and A=2,3,4, find A.For U=7,8,9,10,11 and A=7,11, find A.In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(U)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(A)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(B)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(AB)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(AB)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(B)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(A)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(AB)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(BA)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(AB)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(AB)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(AB)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(AA)In Problems 31-44, refer to the Venn diagram below and find the indicated number of elements. n(AA)If R=1,2,3,4 and T=2,4,6, find AxxRandxTBRT46EFor P=1,2,3,4 and Q=2,4,6, and R=3,4,5,6, find PQR.For P,Q,andR in Problem 47, find PQR.49E50E51E52E53E54E55E56E57E58EIn Problems 59-62, are the given sets disjoint? Let H,T,P,andE denote the sets in Problems 49, 50, 51, and 52, respectively. HandTIn Problems 59-62, are the given sets disjoint? Let H,T,P,andE denote the sets in Problems 49, 50, 51, and 52, respectively. EandPIn Problems 59-62, are the given sets disjoint? Let H,T,P,andE denote the sets in Problems 49, 50, 51, and 52, respectively. PandHIn Problems 59-62, are the given sets disjoint? Let H,T,P,andE denote the sets in Problems 49, 50, 51, and 52, respectively. EandEIn Problems 63-72, discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counterexample. If AB, then AB=A.In Problems 63-72, discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counterexample. If AB, then AB=A.In Problems 63-72, discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counterexample. If AB=A, then AB.In Problems 63-72, discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counterexample. If AB=A, then AB.In Problems 63-72, discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counterexample. If AB=, then A=.In Problems 63-72, discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counterexample. If A=, then AB=.In Problems 63-72, discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counterexample. If AB, then AB.In Problems 63-72, discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counterexample. If AB, then BA.In Problems 63-72, discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counterexample. The empty set is an element of every set.In Problems 63-72, discuss the validity of each statement. Venn diagrams may be helpful. If the statement is true, explain why. If not, give a counterexample. The empty set is a subset of the empty set.How many subsets does each of the following sets contain? AaBa,bCa,b,cDa,b,c,d74E75E76E77E78E79E80E81E82E83E84E85E86E87E88ECommittee selection. A company president and three vice-presidents, denoted by the set P,V1,V2,V3, wish to select a committee be formed? That is, how many 2-person subsets can be formed from a set of 4 people?Voting coalition. The company’s leaders in Problem 89 decide for or against certain measures as follows: The president has 2 votes and each vice-president has 1 vote. Three favorable votes are needed to pass a measure. List all minimal winning coalitions; that is, list all subsets of P,V1,V2,V3 that represent exactly 3 votes.91E92E93E94E95E96E97E98ELet A,B,andC be three sets. Use a Venn diagram (Fig. 3) to explain the following equation: nABC=nA+nB+nCnABnACnBC+nABCThe survey in Example 1 also indicated that 345 firms offer their employees group life insurances, 285 offer long-term disability insurance. How many firms offer their employees group life insurance or long-term disability insurance?Students at a university have the option to stream TV shoes over the internet or to watch cable TV. A survey of 100 college students produced the following results: In the past 30 days, 65 people have streamed TV shows, 45 have watched cable TV, and 30 have done both. A During this 30 -day period, how many people in the survey have streamed TV shows but not watched cable TV? B How many have watched cable TV but not streamed TV shows? C How many have done neither? D Organize this information in a table.A company offers its employees health plans from three different companies: R,S,andT. Each company offers two levels of coverage, AandB, with one level requiring additional employee contributions. What are the combined choices and how many choices are there? Solve using diagram.Each question on a multiple-choice test has 5 choices. If there are 5 such questions on a test, how many different responses are possible for each question?How many 4 -letter code words are possible using the first 10 letters of the alphabet under the three different conditions stated in Example 5 ?In Problems 1-6, Solve for x. (If necessary, review Section 1.1). 50=34+29xIn Problems 1-6, Solve for x. (If necessary, review Section 1.1). 124=73+87xIn Problems 1-6, Solve for x. (If necessary, review Section 1.1). 4x=23+42xIn Problems 1-6, Solve for x. (If necessary, review Section 1.1). 7x=51+45xIn Problems 1-6, Solve for x. (If necessary, review Section 1.1). 3x+11=65+x14In Problems 1-6, Solve for x. (If necessary, review Section 1.1). 12x+5=x+122297E8E9E10EAn entertainment guide recommends 6 restaurants and 3 plays that appeal to a couple. A If the couple goes to dinner or a play, but not both, how many selections are possible? B If the couple goes to dinner and then to a play, how many combined selections are possible?A college offers 2 introductory courses in history, 3 in science, 2 in mathematics, 2 in philosophy, and 1 in English. A If a freshman takes one course in each area during her first semester, how many course selections are possible? B If a part-time student can afford to take only one introductory course, how many selections are possible?How many 3 -letter words can be formed from the letters A,B,C,D,E if no letter is replaced? If letters can be repeated? If adjacent letters must be different?How many 4 -letter code words can be formed from the letters A,B,C,D,E,F,G if no letter is repeated? If letters can be repeated? If adjacent letters must be different?A country park system rates its 20 golf courses in increasing order of difficulty as bronze, silver, or gold. There are only two gold courses and twice as many bronze as silver courses. A If a golfer decides to play a round at a silver or gold course, how many selections are possible? B If a golfer decides to play one round per week for 3 weeks, first on a bronze course, then silver, then gold, how many combined selections are possible?The 14 colleges of interest to a high school senior include 6 that are expensive (tuition more than $30,000 per year), 7 that are far from home (more than 200 miles away), and 2 that are both expensive and far from home. A If the student decides to attend a college that is not expensive and within 200 miles of home, how many selections are possible? B If the student decides to attend a college that is not expensive and within 200 miles of home, during his first two years of college, and then will transfer to a college that is to expensive but far from home, how many selections are possible?